# The Computational Basis of Biological Motion - Pt. 1

How do complex organisms consisting of billions of individual cells achieve smooth, efficient, directed movement? Does evolution favor radically different and increasingly specialized solutions for every biological morphology or is there some common algorithmic basis for movement?

# Galvani, Galvanism, Frankenstein and The Battery

In the mid to late 1700's, interest in scientific circles rode on a wave of excitement surrounding the newly tamed phenomenon of electricity via static generation. One such scientist, an Italian physician and researcher by the name of Luigi Galvani, working at the University of Bologna, became interested in the application of electricity to the body. In his first work on the subject, Commentarius, published in 1791, Galvani described experiments with severed frog legs in which he was able to induce motion by establishing an electrical connection between a nerve fiber and the skin. This lead Galvani to theorize an 'animal electricity' which flowed through the nerve to the outside of the muscle. His theory of electrically induced motion was labelled 'galvanism'.  While incorrect, his mistake would prove just as important as the original discovery and drove Alessandro Volta to invent the first modern battery by storing an electrochemical gradient in an attempt to disprove Galvani.

From Commentarius, 1971. A depiction of Galvani's various experiments in which a metal rod connecting a nerve fiber and the outside of a leg induced motion. Each figure shows varying experimental setups.

As Galvani's experiments began to enter the public consciousness almost nothing was known of the full relationship between electricity and bodily animation. This gap became the subject of imagination and was first popularized by Marry Shelley in Frankenstein - a novel that was heavily influenced by the ideas of galvanism as noted by Shelly in the forward to her book. For most of the 200+ years since Galvani's original discovery science has been unable to yield significant insights into the full computational structure of biological motion, mainly due to the difficulty of constructing appropriate scientific tools for recording or stimulating large numbers of neurons. However, discoveries over the last fifty years have begun to shed light on the basic elements of this structure yielding a picture which is at once fascinatingly complex and yet, in some ways, surprisingly simple.  We will investigate these discoveries into the computational basis of biological motion in vertebrates.

# Motor Neurons

In vertebrates, actuation is driven by spinal microcircuits composed of motor neurons and peripheral connections. Motor neurons typically reside in the spinal column and create tension in muscles by electrically stimulating a chemical reaction in the muscle fibers which causes contraction. The two main types of motor neurons are α-motor neurons and γ-motor neurons, with α's being about twice as prevalent as γ's. Their location in the center of the spinal column conveniently situates motor neurons along the two-way information highway with the brain called the corticospinal tract which passes sensory information to the brain and carries cortical control signals in the opposite direction. The spine not only supports the body mechanically but additionally allows motor neurons to be physically close to muscles and sensors thus reducing the latency around control. For this reason, motor neurons controlling the neck and face are located in the brainstem instead of the spine.

α-MNs are highly connected with sensory feedback such as tendon extension, temperature, and pressure sensitivity. These sensory feedback signals form simple, closed loop controllers that are the basis for reflexes and joint limits. For example, α-MN output is suppressed by tendon extension which prevents the muscle from achieving intentional hyper extension. These unit controllers allow the body to be locally adaptive on a very short timescale without the need to completely route information from the sensory neuron to the brain and back. As a consequence, we can elicit reflex responses such as retracting from a hot surface or kicking when struck underneath the knee without any direct cognition.

α-MNs are myelinated having large, heavily myelinated axons for high speed and reliable signalling. They have a single axon that travels out through a port in the spine to the muscle and makes branching connections with other alpha motor neurons in the spinal core along with connections to descending neuron fibers through the spine from the brainstem.

Simplistic representation of a MN spinal circuit. Sensory signals along with signals from the brain traveling through the corticospinal tract activate motor neurons in the gray matter center of the spine. Command signals from the cortex are computed together with the sensory receptor signal feedback to form a closed loop controller. This interaction forms the most basic computational circuit in biological movement.

In contrast to α-MNs , γ-MNs do not cause the direct activation of muscles but instead act effectively as variable gain units.  They are more lightly myelinated and have slender axons. There are two kinds of γ-MNs, one which is sensitive to the magnitude of muscle stretch and the other which is sensitive to the stretch velocity. When gamma activation is abnormally low, the muscles are too loose and the limb can be easily perturbed, conversely when activation is abnormally high the arm is tense and resistant to movement. Thus, gamma motor neurons serve as a kind of variable impedance controller.

Typically motor neurons are physically grouped in clusters, called 'pools', that share some commonality in controlling a particular muscle or group of muscles. Simple electrical circuits (relatively speaking) composed of alpha neurons, gamma neurons, sensory signals, and brainstem connections are found in all vertebrates and form the lowest level of computation on top of which all motion is achieved.

# Motor Primitives and Central Pattern Generators

Almost all vertebrates consist of many muscle groups actuated by thousands of alpha and gamma motor neurons. No single motor neuron controls an entire muscle and additionally muscle actuation tends to be highly nonlinear. How does the brain manage to create coherent movement out of these spinal microcircuits? A key insight into this control problem was discovered by Emillio Bizzi in the early 1990s through experiments on the frog following closely to the experiments originally done by Galvani.

Unlike Galvani who stimulated severed frog legs with large electrodes on the outside of the skin, Bizzi was able to stimulate motion in a living frog by inserting an electrode directly into the spine that allowed for very specific, local electrical activation. By varying the location and depth of the electrode, Bizzi found that certain regions of the gray matter in the spinal column produced smooth force vector fields at the end effector (Bizzi et al., 1991). These fields are called Convergent Force Fields (CFFs) in the original paper, but the term has been replaced rather universally in the literature by the less restrictive term of 'motor primitive'. Regardless of the name, CFFs had three very important and surprising properties (Mussa-Ivaldi et al., 2000):

• The majority of CFFs converge at a certain final position regardless of the initial position. The final position was the same every time a given CFF was activated.
• CFFs in frogs varied cyclically with time causing a smooth, rhythmic movement in the equilibrium point. Furthermore, the time dependency was decoupled from position suggesting separate time and position pattern generating circuity.
• Most surprisingly, the activation of two CFFs at the same time created a new field that was the linear superposition of the two original fields weighted by the intensity of activation.

These same field generating circuits were found in the spinal columns of cats (Grill, 2004) and turtles (Stein, 2005) suggesting a commonality in control across widely separate branches in the evolutionary tree. In goal directed movement, such as reaching, the equilibrium position does not jump from the current position to the goal but rather is gradually shifted from start to final position along a virtual trajectory (Bizzi et al., 1984).  How the brain constructs these virtual trajectories from a set of CFFs is, to the best of my knowledge, still unknown.

Bizzi constructs a mathematical model of these spinal CFFs by introducing a force field defined as

$$\chi(q,\dot{q}) = K (q_0-q)e^{-(q-q_0)K(q-q_0)} + B \cdot \dot{q}$$

which forms the time independent, spatial component of a CFF (Mussa-Ivaldi et al., 2000) (Aside: There appears to be a typo in the original paper causing the leading term to be as (q-q0) which does not lead to convergence). The B represents natural system damping present in biological mechanics and K is the stiffness. Note that this is essentially a PD controller modulated by an additional exponential term introduced to limit the growth of torques.

The field is then modulated by a spatially independent, rhythmic timing circuit F(t). This circuit does not depend upon the position or velocity feedback information as CFF magnitude varied rhythmically in time even when the leg was clamped down at a specific position. Thus, each CFF is modeled as a combination of a rhythmic modulator and a spatial field also called a pattern generator.

$\Phi_i (t, q, \dot{q}) = F(t)\cdot \chi(q,\dot{q})$

These fields are then activated and summed linearly to form the final control signal output.

$\Lambda \leftarrow \sum_{i=1}^N w_i \Phi_i$

Example $$\Lambda$$ constructed from four spatial fields centered at (5,5), (-5,5), (-5,-5), and (5,-5) are summed linearly according to a cyclical timing function F(t) = sin(t). Negative activation is set to zero. The resulting field forms a local limit cycle that moves the equilibrium point in a square pattern around the origin.

The groups of neurons that give rise to rhythmic CFFs are called Central Pattern Generators, hereafter referred to as CPGs. Although CFFs weren't discovered until the early 1990's, simple models of CPGs were originally theorized by Thomas Graham Brown from the 1960's onward. In vertebrates, CPGs typically consist of separate neuron groups that form a two layer system. On the top layer, a rhythmically active excitatory group of neurons act as a timing clock that periodically activate a lower layer of motor pattern generating neurons which generate convergent fields. The most important feature of the rhythmic generator is that the pattern continues even in the absence of outside signaling (Guertin, 2009). Thus the cortical neurons need only to provide the initial stimulation for the rhythmic generator and possibly provide a signal to modulate the frequency.

Diagram of a two layer CPG. The Rythmic Generator forms excitation connections with various pattern generators and rhythmically cycles through activation. CPGs are found in locomotion tasks such as swimming, walking, and breathing. In simple neuronal structures, the rhythmic generator is connected directly to motor neurons and does not need a pattern generator.

CPGs are the basis for many different kinds of motion such as locomotor movement, breathing, eating and processing of food along the digestive tract. The search and characterization of CPGs is still a very active area of research and for most species CPGs are still a kind of black box. There is no general consensus on the structure of CPGs and a significant amount of evidence suggests it does vary across species morphology. A very good review of the history and current models can be found in Guertin, 2009 and a more detailed treatment in Grillner, 2003. CPGs are the the last fundamental computational unit of biological motion control that can be found across both invertebrates and vertebrates, from snails to fish to cats.

Taken from Guertin, 2009. A schematic representation of the CPG found in the spinal cord based on the most recent evidence found in rodents. The CPG shown here controls the lower limbs.

CPGs are the basis for many different kinds of motion such as locomotor movement, breathing, eating and processing of food along the digestive tract. The search and characterization of CPGs is still a very active area of research and for most species, the CPG is still a kind of black box. There is no general consensus on the structure of CPGs and a significant amount of evidence suggests it does vary across species morphology and across biological purpose. A very good review of the history and current models can be found in Guertin, 2009 and a more detailed treatment of the physiology in Grillner, 2003. CPGs are the the last "fundamental" computational unit of biological motion control that can be found across both invertebrates and vertebrates, from snails to fish to cats.

In vertebrates one additional control component appears to be crucial, called the Mesencephalic Locomotor Region (MLR). Found in the brainstem, the MLR has incoming connections from the sub-cortical regions and descending connections to the spinal tract. The speed of locomotion can be varied depending on the intensity of stimulation to the MLR with low stimulation resulting in low frequency movements and vice versa for high levels of stimulation (Grillner et al., 1997). Direction can be similarly control via asymmetrical stimulation. Amazingly, varying the stimulation to the MLR seems to be able to switch locomotor gait modes in vertebrates such as walking to trot to running in cats as the input increases (Duysens, 1998). Thus, The MLR acts as a kind of interface between brain function and motor output providing a very simple interface of only a few parameters for controlling speed and direction.

The cat in this video demonstrates variation in speed showing smooth transitions between locomotor gates.

In pt. 2, we will look at the higher level, cortical structures involved in motion in mammals such as the cerebellum, the basal ganglia, and the motor cortex.

### References

Bizzi E, Accomero N, Chapple W, Hogan N (1984) Posture control and trajectory formation during arm movement. J Neurosci 4:2738-2744.

Bizzi E, Mussa-Ivaldi FA, Giszter SF (1991) Computations underlying the execution of movement: a novel biological perspective. Science 253:287-29 1.

Duysens J, van de Crommert HWAA. (1998) Neural control of locomotion. Part 1: the central pattern generator from cats to humans (review). Gait Posture 1998;7:131–41

Guertin P. A. (2009) The mammalian central pattern generator for locomotion. Brain Res. Rev. 62, 45–56. doi:10.1016/j.brainresrev.2009.08.002

Grillner S. 2003. The motor infrastructure: from ion channels to neuronal networks.Nat. Rev.Neurosci. 4:573–86

Grillner S., Georgopolulos, A.P., & Jordan, L. M. (1997). Selection and initiation of motor behavior. In P.S.G. Stein, S. Grillner, A. Selverston, & D. G. Stuart (eds.), Neurons, networks and motor behavior. MIT Press

Lemay MA, Grill WM (2004) Modularity of motor output evoked by intraspinal microstimulation in cats. J Neurophysiol 91:502–514.

Mussa-Ivaldi FA, Bizzi E (2000) Motor learning through the combination of primitives. Philos Trans R Soc Lond B Biol Sci 355: 1755–1769.

Stein PS. (2005). Neuronal control of turtle hindlimb motor rhythms. J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 191:213–29